Superfluids and their vortices

Subir Sachdev

Talk online:http://pantheon.yale.edu/~subir

Superfluidity/superconductivity occur in:

• liquid 4He• metals Hg, Al, Pb, Nb,

Nb3Sn…..• liquid 3He• neutron stars• cuprates La2-xSrxCuO4,

YBa2Cu3O6+y….• M3C60• ultracold trapped atoms• MgB2

The reason for superflow:The Bose-Einstein condensate:

A macroscopic number of bosons occupy the lowest energy quantum state

Such a condensate also forms in systems of fermions, where the bosons are Cooper pairs of fermions:

yk

xk ( )( )2 2

Pair wavefunction in cupra

s:

te

x yk kΨ = − ↑↓ − ↓↑

0S =

Velocity distribution function of ultracold 87Rb atoms

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wiemanand E. A. Cornell, Science 269, 198 (1995)

87Rb bosonic atoms in a magnetic trap and an optical lattice potential

M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).

The strength of the period potential can be varied in the experiment

Strong periodic potential“Eggs in an egg carton”

Tunneling between neighboring minima is negligible and atoms remain localized in a well. However, the total wavefunction must be symmetric between exchange

Strong periodic potential“Eggs in an egg carton”

Tunneling between neighboring minima is negligible and atoms remain localized in a well. However, the total wavefunction must be symmetric between exchange

Strong periodic potential“Eggs in an egg carton”

Tunneling between neighboring minima is negligible and atoms remain localized in a well. However, the total wavefunction must be symmetric between exchange

= + +

Insulator

+ + +

○ ○ ○

○

○ ○ ○

○ ○

○ ○ ○

○ ○ ○ ○ ○ ○

Weak periodic potential

A single atom can tunnel easily between neighboring minima

Weak periodic potential

○A single atom can tunnel easily between

neighboring minima

Weak periodic potential

○

A single atom can tunnel easily between neighboring minima

Weak periodic potential

○

A single atom can tunnel easily between neighboring minima

Weak periodic potential

( ) = + G + ie φ ○ ○ ○

The ground state of a single particle is a zero momentum state, which is a quantum superposition of

states with different particle locations.

The Bose-Einstein condensate in a weak periodic potentialLowest energy state for many atoms

3

=

= + + +

BE

+ + + + ....27 t

C G

er

G G

ms

()

ie φ ○○ ○ ○

○○

○ ○○

○

○

○○ ○

○

○○

○○○ ○

Large fluctuations in number of atoms in each potential well – superfluidity (atoms can “flow” without dissipation)

The Bose-Einstein condensate in a weak periodic potentialLowest energy state for many atoms

3

=

= + + +

BE

+ + + + ....27 t

C G

er

G G

ms

()

ie φ ○○ ○ ○

○○

○ ○○

○

○

○○ ○

○

○○

○○○ ○

Large fluctuations in number of atoms in each potential well – superfluidity (atoms can “flow” without dissipation)

The Bose-Einstein condensate in a weak periodic potentialLowest energy state for many atoms

3

=

= + + +

BE

+ + + + ....27 t

C G

er

G G

ms

()

ie φ ○○ ○ ○

○○

○ ○○

○

○

○○ ○

○

○○

○○○ ○

Large fluctuations in number of atoms in each potential well – superfluidity (atoms can “flow” without dissipation)

The Bose-Einstein condensate in a weak periodic potentialLowest energy state for many atoms

3

=

= + + +

BE

+ + + + ....27 t

C G

er

G G

ms

()

ie φ ○○ ○ ○

○○

○ ○○

○

○

○○ ○

○

○○

○○○ ○

Large fluctuations in number of atoms in each potential well – superfluidity (atoms can “flow” without dissipation)

The Bose-Einstein condensate in a weak periodic potentialLowest energy state for many atoms

3

=

= + + +

BE

+ + + + ....27 t

C G

er

G G

ms

()

ie φ ○○ ○ ○

○○

○ ○○

○

○

○○ ○

○

○○

○○○ ○

Large fluctuations in number of atoms in each potential well – superfluidity (atoms can “flow” without dissipation)

The Bose-Einstein condensate in a weak periodic potentialLowest energy state for many atoms

3

=

= + + +

BE

+ + + + ....27 t

C G

er

G G

ms

()

ie φ ○○ ○ ○

○○

○ ○○

○

○

○○ ○

○

○○

○○○ ○

Large fluctuations in number of atoms in each potential well – superfluidity (atoms can “flow” without dissipation)

The Bose-Einstein condensate in a weak periodic potentialLowest energy state for many atoms

3

=

= + + +

BE

+ + + + ....27 t

C G

er

G G

ms

()

ie φ ○○ ○ ○

○○

○ ○○

○

○

○○ ○

○

○○

○○○ ○

Large fluctuations in number of atoms in each potential well – superfluidity (atoms can “flow” without dissipation)

87Rb bosonic atoms in a magnetic trap and an optical lattice potential

M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).

The strength of the period potential can be varied in the experiment

Superfluid-insulator quantum phase transition at T=0

M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).

Understanding superflow

( )

( ) ( )( ) ( )

1 2 3

1 2 3

1 2 3

The wavefunction of bosons at rest

, , ,The wavefunction of bosons flowing with velocity

, , ,

=

s N

N

s

mi

N

i i

e

e θ θ

• + +

+

Ψ

Ψ…

…

…1 2

v r +r +r r

r r

r r r rv

r r r r( ) ( ) ( )1 2 3, , ,

where

(for non-Galilean invariant superfluids, the co-efficient of

i iN

s m

θ θ

θ

θ

+ + +

∇

Ψ

=

… …3 Nr r r r r r

v ∇

is modified)

Understanding superflowPersistent currents

2C

dx nθ π∇ =∫No local change of the

wavefunction can change the value of n

Supercurrent flows “forever”

C

Vortices in the superfluid

Magnus forces, duality, and point vortices as dual “electric” charges

Excitations of the superfluid: Vortices

Observation of quantized vortices in rotating 4He

E.J. Yarmchuk, M.J.V. Gordon, and R.E. Packard,

Observation of Stationary VortexArrays in Rotating Superfluid Helium, Phys. Rev. Lett. 43, 214 (1979).

Observation of quantized vortices in rotating ultracold Na

J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle, Observation of Vortex Lattices in Bose-Einstein Condensates, Science 292, 476 (2001).

Quantized fluxoids in YBa2Cu3O6+y

J. C. Wynn, D. A. Bonn, B.W. Gardner, Yu-Ju Lin, Ruixing Liang, W. N. Hardy, J. R. Kirtley, and K. A. Moler, Phys. Rev. Lett. 87, 197002 (2001).

Excitations of the superfluid: Vortices

Central question:In two dimensions, we can view the vortices as

point particle excitations of the superfluid. What is the quantum mechanics of these “particles” ?

In ordinary fluids, vortices experience the Magnus Force

FM

( ) ( ) ( )mass density of air velocity of ball circulationMF = i i

Dual picture:The vortex is a quantum particle with dual “electric”

charge n, moving in a dual “magnetic” field of strength = h×(number density of Bose particles)

87Rb bosonic atoms in a magnetic trap and an optical lattice potential

M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).

The strength of the period potential can be varied in the experiment

Upon approaching the insulator, the phase of the condensate becomes “uncertain”.

Vortices cost less energy and vortex-anti-vortex pairs proliferate.

The quantum mechanics of vortices plays a central role in the superfluid-insulator

quantum phase transition.